Contents

  1. Part I: RJMCMC - \(exp(1)\) prior on \(\beta\)
  1. Data
  1. RJMCMC
  1. Part II: RJMCMC - \(\Gamma(2, 1/2)\) prior on \(\beta\)

Background

RJMCMC was implemented whereby the algorithm jumped between the Baseline model \(M_1\) of an epidemic with regular spreading events \((\alpha)\) and the Super Spreading Events (SSE) model \((\alpha, \beta, \gamma)\) which has both regular spreading events with rate \(\alpha\) and super spreading events with rate \(\beta\) and multiplicative factor \(\gamma\).

Bayes Factor

Bayesian model comparison is a method of model selection based on Bayes factors. The aim of the Bayes factor is to quantify the support for one model over another, e.g model \(M_1\) over model \(M_2\). The Bayes Factor BF is as follows;
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)P(D)}{P(M_1)}}{\dfrac{P(M_2|D)P(D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_2 | D)} \]
when \(P(M_1) == P(M_2)\), otherwise
\[ BF = \dfrac{P(D|M_1)}{P(D|M_2)} = \dfrac{\dfrac{P(M_1|D)}{P(M_1)}}{\dfrac{P(M_2|D)}{P(M_2)}} = \dfrac{P(M_1|D)}{P(M_1)} \cdot \dfrac{P(M_2)}{P(M_2|D)} \]
where \(P(D|M_1)\) is the model evidence, specifically the marginal likelihood integrand;
\[ P(D|M_1) = \int P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta) \hspace{1 mm} P(\hspace{1 mm}\theta \hspace{1 mm}| M_1) \hspace{1 mm}d \theta\] and the first term in the integrand \(P(D \hspace{1 mm}|\hspace{1 mm} M_1, \theta)\) is the likelihood and the second term \(P(\hspace{1 mm}\theta \hspace{1 mm}| M_1)\) is the prior on the model parameter \(\theta\).



Interpretation of the Bayes Factor results


A Bayes Factor > 1 signifies that \(M_1\) is more strongly supported by the data under consideration than \(M_2\). Harold Jefferys gave a scale of interpretation of the Bayes Factor;

Bayes Factor Bayes Factor equivalence Evidence Strength
< 10^0 < 1 Negative (supports M_2)
[10^0, 10^1/2] [1, 3.16] Weak evidence
[10^1/2, 10^1] [3.16, 10] Substantial
[10^1, 10^3/2] [10, 31.62] Strong
[10^3/2, 10^2] [31.62, 100] Very strong
> 10^2 > 100 Decisive


Part I: RJMCMC between Base model & SSE Model - exp(1) prior used for beta

An \(exp\hspace{1mm}(\beta; \hspace{1mm}1)\) has density;

Model Comparison - SSE data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.31  0.1 0.01    10 0.19 1.8  0.25         70.85   86.68
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        983    12.7        144     99.56         1129          876
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          258     77.25          875          181     82.86   0.8866
##   beta_pc_non_0    bf
## 1        0.1134 7.818


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.63  0.1 0.07    10 5.49 1.8  1.06          2.16    6.28
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        628    2.56        256      6.37          637            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.45  0.1 0.16    10 6.62 1.8  1.46          4.71    5.18
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        518    1.91        191      5.42          542            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.77  0.1 0.02    10 0.26 1.8   0.8         10.32    84.7
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1306   14.59        225        93         1434         1464
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           78     94.94         1464         1735     45.76   0.8457
##   beta_pc_non_0    bf
## 1        0.1543 5.481


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.75  0.1 0.02    10 0.18 1.8  1.77          3.63   53.78
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        675    8.69        109     43.35          544         1244
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           11     99.12         1243         4288     22.47   0.8745
##   beta_pc_non_0    bf
## 1        0.1255 6.968


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.74  0.1 0.08    10 0.43 1.8  1.85          9.88    62.6
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1734    9.31        258     39.93         1106         2584
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          186     93.29         2584         1964     56.82   0.7229
##   beta_pc_non_0    bf
## 1        0.2771 2.609

Part 2: Gamma Prior on Beta

A gamma prior, \(\Gamma(\beta; k, \theta)\) on beta was also trialed whereby \(k\) determines the shape of the distribution and \(\theta\) governs the scale. The gamma distribution function is as follows;

\[ \Gamma(\beta; k, \theta) = \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \]

A range of gamma priors on beta are used including a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2.5, \hspace{1mm} 2)\) and a \(\Gamma \hspace{1mm}(\beta; \hspace{1mm} 3, \hspace{1mm} 2)\) Each have a mean of \(k \cdot \theta\). A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}2, \hspace{1mm} 2.5)\), is as follows;


A \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 2)\);



And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}3, \hspace{1mm} 3)\);

And a \(\Gamma\hspace{1mm}(\beta; \hspace{1mm}4, \hspace{1mm} 4)\);

In the Metropolis acceptance step, the logs of all quantities are determined and evaluating \(log\hspace{1mm}( \Gamma\hspace{1mm}(\beta; \hspace{1mm} k, \hspace{1mm} \theta))\) gives;


\[log\bigg( \dfrac{1}{\Gamma(k) \cdot \theta^k}\cdot \beta^{(k-1)} \cdot e^{\dfrac{-\beta}{\theta}} \bigg)\]

\[ = \dfrac{1}{log\Gamma(k)\cdot klog(\theta)} \cdot (k-1) \cdot log(\beta) \cdot \dfrac{-\beta}{\theta} \]


Gamma Prior \(\Gamma\)(\(\beta\); 2, 2.5)


Model Comparison - SSE data I


##   rep n_mcmc alpha a_mc beta b_mc gamma   g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.48  0.1 0.09    10 307.09 1.8 25.42         78.43    0.01
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          1    99.5       9856      0.03            3           38
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9868      0.38           38            4     90.48   0.0093
##   beta_pc_non_0    bf
## 1        0.9907 0.009


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.58  0.1  0.1    10 19.46 1.8  2.53         42.64       0
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          0   60.29       6028         0            0            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.62  0.1  0.1    10 13.22 1.8  1.94         32.09       0
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          0   15.51       1551         0            0            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 1.02  0.1 0.08    10  3.8 1.8  1.52         22.89    0.18
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          6   52.78       1729      1.01           33         1938
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         1338     59.16         1938         1113     63.52   0.6723
##   beta_pc_non_0    bf
## 1        0.3277 2.052


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.77  0.1 0.02    10 0.18 1.8  1.79          3.38    0.08
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          1   18.77        234      0.56            7         1246
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1            1     99.92         1246         4299     22.47   0.8752
##   beta_pc_non_0    bf
## 1        0.1248 7.013


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.74  0.1 0.09    10 0.48 1.8  1.87          9.94    0.28
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1          8   16.86        488       2.8           81         2619
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          275      90.5         2619         1885     58.15   0.7105
##   beta_pc_non_0    bf
## 1        0.2895 2.454


Gamma Prior \(\Gamma(\beta; 3, 2)\)


Model Comparison - SSE data I


##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.54  0.1 0.08    10    9 1.8  2.18         79.71   89.08
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       3849   83.89       3625     87.06         3762         1285
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         3036     29.74         1285          352      78.5   0.5678
##   beta_pc_non_0    bf
## 1        0.4322 1.314


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.63  0.1 0.09    10 46.62 1.8  3.41         36.21   31.74
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       3174   61.09       6108     53.19         5318            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.42  0.1 0.19    10 10.81 1.8  2.32         15.31     9.4
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        940    9.34        934     13.58         1358            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.95  0.1 0.05    10 0.82 1.8  1.06         17.04   82.39
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       2045   48.23       1197     81.91         2033         1766
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          716     71.15         1765         1398      55.8   0.7518
##   beta_pc_non_0    bf
## 1        0.2482 3.029


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.77  0.1 0.02    10 0.19 1.8  1.79          3.74   53.91
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        676   17.38        218     36.44          457         1238
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           16     98.72         1237         4343     22.17   0.8746
##   beta_pc_non_0    bf
## 1        0.1254 6.974


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.75  0.1 0.08    10 0.49 1.8  1.87         10.31   61.59
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1770   16.84        484     34.45          990         2619
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          255     91.13         2618         1888      58.1   0.7126
##   beta_pc_non_0    bf
## 1        0.2874 2.479


Gamma Prior \(\Gamma(\beta; 3, 3)\)


Model Comparison - SSE data I


##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.54  0.1 0.08    10    9 1.8  2.18         79.71   89.08
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       3849   83.89       3625     87.06         3762         1285
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         3036     29.74         1285          352      78.5   0.5678
##   beta_pc_non_0    bf
## 1        0.4322 1.314


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.63  0.1 0.09    10 46.62 1.8  3.41         36.21   31.74
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       3174   61.09       6108     53.19         5318            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 0.42  0.1 0.19    10 10.81 1.8  2.32         15.31     9.4
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        940    9.34        934     13.58         1358            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8 0.95  0.1 0.05    10 0.82 1.8  1.06         17.04   82.39
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       2045   48.23       1197     81.91         2033         1766
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          716     71.15         1765         1398      55.8   0.7518
##   beta_pc_non_0    bf
## 1        0.2482 3.029


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.77  0.1 0.02    10 0.19 1.8  1.79          3.74   53.91
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        676   17.38        218     36.44          457         1238
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           16     98.72         1237         4343     22.17   0.8746
##   beta_pc_non_0    bf
## 1        0.1254 6.974


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.75  0.1 0.08    10 0.49 1.8  1.87         10.31   61.59
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1770   16.84        484     34.45          990         2619
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          255     91.13         2618         1888      58.1   0.7126
##   beta_pc_non_0    bf
## 1        0.2874 2.479


Gamma Prior \(\Gamma(\beta; 4, 4)\)


Model Comparison - SSE data I

##   rep n_mcmc alpha a_mc beta b_mc gamma  g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.54  0.1 0.09    10 13.97 1.8  3.08         79.43   87.93
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       3862   85.93       3774     85.77         3767         1174
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         3218     26.73         1174          284     80.52   0.5607
##   beta_pc_non_0    bf
## 1        0.4393 1.276


Model Comparison - SSE data II

##   rep n_mcmc alpha a_mc beta b_mc gamma   g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 0.67  0.1 0.07    10 115.87 1.8  5.09         36.44   30.25
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       3025   65.45       6544     58.15         5814            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data III

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8  0.4  0.1 0.19    10 10.5 1.8  2.27         16.22    9.85
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        985    8.77        877     14.09         1409            0
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1         9999         0            0            0       NaN        0
##   beta_pc_non_0 bf
## 1             1  0


Model Comparison - SSE data IV

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   3  10000   0.8  0.9  0.1 0.04    10 0.85 1.8  1.02         16.68   84.78
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       2005   49.73       1176     82.88         1960         1716
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          649     72.56         1715         1414     54.81   0.7635
##   beta_pc_non_0    bf
## 1        0.2365 3.228


Model Comparison - BASE Data I

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   1  10000   0.8 1.77  0.1 0.02    10 0.19 1.8  1.79          3.75   54.83
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1        687   17.72        222     36.39          456         1238
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1           15      98.8         1238         4308     22.32   0.8746
##   beta_pc_non_0    bf
## 1        0.1254 6.974


Model Comparison - BASE Data II

##   rep n_mcmc alpha a_mc beta b_mc gamma g_mc  R0 R0_mc accept_rate_a a_rte_b
## 1   2  10000   0.8 1.75  0.1 0.09    10 0.54 1.8   1.9          10.1   59.74
##   n_accept_b a_rte_g n_accept_g a_rte_b_g n_accept_b_g n_accept_rj0
## 1       1751   16.41        481     32.82          962         2596
##   n_reject_rj0 a_rte_rj0 n_accept_rj1 n_reject_rj1 a_rte_rj1 beta_pc0
## 1          335     88.57         2595         1889     57.87   0.7069
##   beta_pc_non_0    bf
## 1        0.2931 2.412


Bayes Factor - Summary of Results

In this particular setting in which the Baseline model \(\alpha\) and SSE model \((\alpha, \beta, \gamma)\) are compared, the Bayes factor is calculated as;

\[ \dfrac{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples == 0}{Proportion \hspace{1 mm}of \hspace{1 mm} \beta \hspace{1 mm} mcmc \hspace{1 mm} samples != 0} \]


The following table summaries the results of the RJMCMC iterations for a number of datasets when both a \(exp(\beta, 1)\) prior and a variety of \(\Gamma(\beta;)\) priors on beta were used.



Epidemic Data Max daily infection count Bayes Factor exp(beta; 1) Bayes Factor,gamma(2, 2.5) Bayes Factor,gamma(3, 2) Bayes Factor,gamma(3, 3) Bayes Factor,gamma(4, 4)
SSE - dies out 2 7.818 0.009 1.314 1.314 1.276
SSE spreads 55 0 0 0 0 0
SS spreads 340 0 0 0 0 0
SSE - dies out 2 5.481 2.052 3.029 3.029 3.228
Base - spreads 65 6.968 7.013 6.974 6.974 6.974
Base - spreads 45 2.609 2.454 2.479 2.479 2.412